2. Scope Introduction Local radial basis function collocation method (LRBFCM) The adaptation algorithm The refinement criteria The refinement algorithm Numerical examples –2D diffusion equation –Convection dominated trans port in circular velocity field Conclusions and further work

3. Overview of our publications - RBFCM I.Kovačević, A.Poredoš, B.Šarler, Numerical heat Transfer B (2003) (Stefan problem) B.Šarler, J.Perko, D.Gobin, B.Goyeau, H.Power, Int.J.Numer.Methods Heat & Fluid Flow (2004) (Porous media flow – Darcy model) B.Šarler, Engineering Analysis with Boundary Elements (2004) (Porous media flow – Darcy-Brinkman model) B. Šarler, Computer Modeling in Engineering and Sciences (2005) (Navier Stokes) Problems due to ill-conditioning of the collocation matrix for more than approx. 1000 nodes. Introduction

4. Overview of our recent publications - LRBFCM B.Šarler, R.Vertnik, Computers & Mathematics with Applications (2006) (Diffusion) R.Vertnik, B.Šarler, Int.J.Numer.Methods Heat & Fluid Flow (2006) (Convection-diffusion & phase change) R.Vertnik, M.Založnik, B.Šarler, Eng.Anal.Bound.Elem. (2006) (Continuous casting of aluminium) I.Kovačević, B.Šarler, Materials Science and Engineering A (2006) (Phase field) G.Kosec, B.Šarler, Computer Modeling in Engineering and Sciences (2008) (Navier Stokes) G.Kosec, B.Šarler, J.Numer.Methods Heat & Fluid Flow (2008) (Porous media flow) R.Vertnik, B.Šarler, Cast Metals Research (2008) (Continuous casting of steel) G.Kosec, B.Šarler, Cast Metals Research (2008) (Freezing of pure metals) R.Vertnik, B.Šarler, Computer Modelig in Engineering and Sciences (2009) (Turbulent flow, k-epsilon model) Method improved and made available in solution of practical problems with the local formulation-LRBFCM. Solutions on several thousand nodes tested. Introduction

5. Solution of thermo-fluid problems with local approach –Local Radial Basis Function Collocation Method –Local pressure velocity coupling presented at FEF 2007 ∙Classical De Wahl Davis benchmark (Kosec, Šarler, 2008) (FEF2007) ∙Natural convection in porous media (Kosec, Šarler, 2008) ∙Meshless Approach to Solving Freezing with Natural Convection (Kosec, Šarler, in review) ∙Convection Driven Melting of Anisotropic Metals (Kosec, Šarler, in print) ∙Melting driven by natural convection (Kosec, Šarler, to be submitted) Extension of the method to the dynamic node configuration –Present work –Other’s research in the field (Libre, 2008, Bozzini, 2002, Gomez, 2006, Sarra, 2005, Schaback, 2000) Principal incitement –How to make LRBFCM h-adaptive in a most simple and effective way?

6. Local Radial Basis Functions Collocation Method Basis functions: Hardy’s multiquadrics LRBFCM RBF basis augmentation in order to improve the method, various functions might be included in the collocation basis. Constant and linear functions are added in the present work in order to enhance the method at the small sub- domain nodal distances regime. - Shape parameter - Spatial coordinate

7. The performance of the LRBFCM One-dimensional analysis is performed to asses the method’s behavior –The exponential function is taken as a test function Discretisation errors of the first and second derivatives are analyzed With respect to –The sub-domain nodal distances –The shape parameter C –Comparison of augmented to non-augmented LRBFCM - RBFCM representation - Exact values

8. Discretisation error of the derivatives. With respect to the nodal distance and shape parameter for non-augmented LRBFCM

9. Second derivative error. Comparison of non-augmented to the augmented LRBCM

10. The adaptation criteria An efficient criteria for adding and/or removing of the nodes is required –Derivatives calculation error extrapolation –Error predictor based on the shape of the field and/or properties of the numerical method The LRBFCM predictor –Based on the norm of LRBFCM collocation coefficients –No need for additional computation –Behaviour with respect to the spatial sub-domain configuration is similar to the behaviour of the derivatives error –Small number of calculation operations (fast) –Robust and very simple to implement Number of sub-domain nodes LRBCM coefficients

11. Error prediction. Comparison of error prediction to the actual derivative error

12. The adaptation algorithm rules Node insertion rules –If P is above the limit value Pmax the node is requested to expand (construct new nodes around it) –Level difference with it’s neighbors after the expansion is not allowed to exceed one –New node (child) can be constructed only when none of the neighbors has it’s child on the same coordinate –All child nodes are positioned symmetrical around parent node in order to retain equidistant sub-domain for all adaptation levels Node removal rules –If P is below the limit value Pmin the node is requested to close (remove child nodes) –Level difference with it’s neighbors after closing is not allowed to exceed one –Child node is allowed to be removed only if none of the neighbor parent node is pointing to it

13. The sub domain configuration Two types of the sub-domains appear (normal/rotated) Five nodded sub-domain The neighbor's neighbor is not necessarily the same node All sub-domains are equidistant

14. The refinement algorithm

15. Node expansion example Initial nodes distribution

16. Node expansion example Level one

17. Node expansion example Level two

18. Node expansion example Level three

19. Node expansion example Level four

20. Node expansion example Level five

21. Node expansion example Level six

22. First numerical example A two-dimensional time-dependent diffusion equation with Dirichlet boundary conditions All values are dimensionless Analytical solution

23. Test cases in the first numerical example Common settings Diffusivity is set to one Domain size is 1x1 Boundary conditions and initial state are the same for all cases All boundaries are set to zero Initial state is set to one Explicit time stepping with time step dt=1e-5 Case definitions Four different cases with different average nodes count through whole simulation are considered Each case with the adaptive node distribution (A) is compared with the case exhibiting the fixed node distribution (S) with the same average number of the nodes All values are dimensionless CaseAverage number of the nodes Case1-1225 Case1-2441 Case1-3625 Case1-4961

24. First numerical example results Average error (left) and number of the nodes (right) with respect to time for cases 1-1 and 1-2.

25. First numerical example results Average error (left) and number of the nodes (right) with respect to time for cases 1-3 and 1-4.

26. Cross-section error comparison

28. Time evolution of the first numerical case

29. Second numerical example Convection dominated transport Neumann boundary conditions on all walls Adiabatic walls Initial state set to zero except on a small region where the initial state is one Circular velocity field High Peclet number (L – domain length) Results with velocity v after one revolution are compared with the results where the velocity is set to zero. The difference is introduced as a deviation Where and stands for simulation time (2π), temperature computed without advection term, temperature computed with the advection term.

30. Common settings Diffusivity is set to 2e-4 Domain size is 2x2 Boundary conditions and initial state are for all cases the same Explicit time stepping with time step dt=1e-5 Angular velocity is 1 and simulation time is 2π Case definitions Four different cases with different average nodes count through whole simulation Each case with adaptive node distribution (A) is compared to the case with static node distribution (S) with the same average nodes count. Test cases for the second numerical example CaseAverage number of the nodes Case2-13600 Case2-26400 Case2-310000 Case2-414400 All values are dimensionless

31. Second numerical example results – case 1

32. Second numerical example results – case 2

33. Second numerical example results – case 3

34. Second numerical example results – case 4

35. Time evolution of the second numerical case

36. Comparison – adaptive vs. static node distribution Adaptive nodes distribution Static nodes distribution

37. Conclusions The developed algorithm is ready to be implemented in more complex physical systems Fluid mechanics with iterative pressure velocity coupling Extension to the multi-phase systems (phase field, level set formulations) Melting and freezing Solidification of a binary system... Issues that need further clarification: –Behavior of the LRBFCM near the nodal density jumps –Rotated domain inclusion effects –Setting the free parameter –Sub-domain size influence –Sub-domain shape influence –Multi-grid behavior

38. Further development Implementation of the developed adaptive numerical approach with completely local pressure velocity coupling in more complex systems The local fluid flow computational approach has been extensively tested with good agreement with other authors There are some difficulties in implementing it with the developed adaptation algorithm

39. References and thank you for your attention [1]Atluri, S.N. and Shen, S. (2002), The Meshless Method, Tech Science Press, Encino. [2] Atluri, S.N. (2004), The Meshless Method (MLPG) for Domain and BIE Discretization, Tech Science Press, Forsyth. [3] Liu, G.R. (2003), Mesh Free Methods, CRC Press, Boca Raton. [4] Liu, G.R. and Gu, Y.T. (2005), An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht. [6] Buhmann, M.D. (2000), Radial Basis Functions, Cambridge University Press, Cambridge. [7] Franke, J. (1982), "Scattered data interpolation: tests of some methods", Mathematics of Computation, Vol. 48, pp. 181- 200. [8]Kansa, E. J. (1990a), "Multiquadrics – a scattered data approximation scheme with application to computational fluid dynamics, part I", Computers and Mathematics with Applications, Vol. 19, pp. 127-145. [9] Kansa, E. J. (1990b), "Multiquadrics – a scattered data approximation scheme with application to computational fluid dynamics, part II", Computers and Mathematics with Applications, Vol. 19, pp. 147-161 [10]Zerroukat, M., Power, H. and Chen, C.S. (1998), "A numerical method for heat transfer problems using collocation and radial basis functions", Internation Journal of Numerical Methods in Engineering, Vol. 42, pp. 1263-1278. [11] Šarler, B., Perko, J., Chen, C. S. and Kuhn, G. (2001), "A meshless approach to natural convection", International Conference on Computational Engineering and Sciences, CD proceedings. [12] Šarler, B. (2005), "A radial basis function collocation approach in computational fluid dynamics", Computer Modelling in Engineering & Sciences, Vol. 7, pp. 185-193. [13] Šarler, B. and Vertnik, R. (2006), "Meshfree explicit local radial basis function collocation method for diffusion problems", Computers and Mathematics with Applications, Vol. 51, pp. 1269-1282. [14] Vertnik, R. and Šarler, B. (2006), "Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems", International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 16, pp. 617-640. [15] Vertnik, R., Založnik, M. and Šarler, B. (2006), "Solution of transient direct-chill aluminium billet casting problem with simultaneous material and Interphase moving boundaries by a meshless method", Engineering Analysis with Boundary Elements, Vol. 30, pp. 847-855. [16] Kovačević, I. and Šarler, B. (2005), "Solution of a phase-field model for dissolution of primary particles in binary aluminum alloys by an r-adaptive mesh-free method", Materials Science and Engineering A, Vol. 413–414, pp. 423–428. [17] Divo, E. and Kassab, A. J. (2007), "An efficient localized RBF meshless method for fluid flow and conjugate heat transfer", ASME Journal of Heat Transfer, Vol. 129, pp. 124-136. [18] Chantasiriwan, S. (2006), "Performance of Multiquadric Collocation Method in Solving Lid-driven Cavity Flow Problem with Low Reynolds Number ", CMES, Vol. 15, pp. 137-146. [19] Kosec, G. and Šarler, B. (2007), "Solution of Thermo-Fluid Problems by Collocation with Local Pressure Correction", International Journal of Numerical Methods for Heat and Fluid Flow, (in press). [20] Kosec, G. and Šarler, B. (2008), "Local RBF collocation method for Darcy flow", CMES, Vol. 25, pp. 197-208. [21] Zhang, Y., Sim, T., Tan, C. L. and Sung, E. (2005), "Anatomy-based face reconstruction for animation using multi-layer deformation", Journal of Visual Languages and Computing, (in press). [22] Berger, M. J. (1983), "Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations", Journal of Computational Physics, Vol. 53, pp. 484-512. [23] Celik, I. and Hu, G. (2004), "Single Grid Error Estimation Using Error Transport Equation", ASME, Vol. 126, pp. 778.

40. Numerical error of the divative With respect to the sub-domain shape The LRBFCM achieves best performance with the equidistant sub domain (R=1)

41. Numerical error of the derivative With respect to curvature of the function